Bäcklund Transformations and Darboux Integrability for Nonlinear Wave Equations

Abstract

Abstract. We prove that second-order Monge-Ampère equations for one function of two variables are connected to the wave equation by a Bäcklund transformation if and only if they are integrable by the method of Darboux at second order.

Symmetry Reduction of Exterior Differential Systems and Bäcklund Transformations for PDE in the Plane

“…5.2) giving an EDSĪ 2 on a five manifold. The EDSĪ 2 is not a Pfaffian system but is known as a Monge-Ampére system [7]. In this case diagram (2.2) can be refined.…”

“…In [2] it is shown that if F = 0 is Darboux integrable (without prolongation) and admits a Bäcklund transformation to the wave equation z xy = 0 (with one-dimensional fibres as in [7]), then the Bäcklund transformation can be constructed by group reduction using Theorem 1. In order to describe these results formally let I 2 be the standard rank 3 hyperbolic Pfaffian system determined by the restriction of the contact system on J 2 (R 2 , R) [9] to the level set M in (2.1).…”

“…The description of Bäcklund transformations leading to diagram (1.4) admits a coordinate invariant description in terms of exterior differential systems (EDS) which we now describe [7]. Two exterior differential systems (EDS) I 1 To explain how group reduction can be used to produce diagram (1.5) we begin with the general definition of reduction of an EDS by a submersion.…”

Geometry of Bäcklund transformations II: Monge–Ampère invariants

“…If locally either V 3 or W 3 is zero, which is independent of the choice of 1-refined liftings, the underlying Bäcklund transformation must relate a hyperbolic Euler-Lagrange system with the system corresponding to z xy " 0. See [CI09] and [Zvy91] for a classification of all hyperbolic Monge-Ampère systems that are rank-1 Bäcklund-related to the equation z xy " 0.…”

Geometry of Bäcklund transformations I: generality

“…In addition to the elliptic and hyperbolic examples mentioned above, a Bäcklund transformation may exist relating solutions of a parabolic equation (see [NC82]) or an equation of order higher than 2, for example, the KdV equation (see [WE73]). Furthermore, two PDE systems being Bäcklund-related need not be contact equivalent to each other (see [CI09]).…”